Lattice Hulls of Euclidean Balls Need Not Be Normal: Counterexamples in Every Dimension at Least Six

Michałek asked in 2017 (Oberwolfach Report 44/2017, Conjecture 1) whether the lattice hull of a Euclidean ball \(B\subset\mathbb{R}^n\), the convex hull of the lattice points in \(B\), is always a normal polytope, and whether there is a very ample, non-normal simple polytope, or even simplex. We answer the first question in the negative in every dimension \(n\ge6\). The convex hull \(P\) of the points \(z\in\mathbb{Z}^6\) with \(|z|^2\)\({}-z_3\)\({}-z_5\)\({}\le158906\), the lattice hull of the ball with centre \((0,0,\frac12,0,\frac12,0)\) and squared radius \(\frac{317813}{2}\), is a spanning lattice polytope that is neither normal nor very ample. For \(\mathbf{n}\)\({}=(-36,-46,50,-57,-41,1)\), the Cauchy–Schwarz inequality gives \(\mathbf{n}\)\({}\cdot z\)\({}\ge\)\({}-41504\) on the lattice points of \(P\); exactly six of them lie on the hyperplane \(\mathbf{n}\)\({}\cdot z\)\({}=-41504\) (a computer-assisted finite enumeration, done by three independent methods); they span an empty simplex facet whose vertices generate a sublattice of index \(2\); and half their sum is a lattice point of \(3P\) that is not a sum of three lattice points of \(P\). Padding with zeros gives examples in every dimension \(n\ge6\); further certificates are given, one in dimension \(7\). For ellipsoids, Gubeladze (2021) gave non-normal examples in every dimension \(d\ge6\); for \(n\le2\) the answer is positive by Pick's theorem, and for \(n=3\) by a theorem of Bruns, Gubeladze and Michałek (2016). No answer for round balls was found in the literature accessible to us. Dimensions \(4\) and \(5\), and balls centred at lattice points, remain open; the face mechanism used here cannot work in dimensions at most \(5\). The second question was answered by an example of Cox, Haase, Hibi and Higashitani (2014), the very ample, non-normal simplex \(3\operatorname{conv}(0,e_1,\)\(\dots,e_6,(3,3,3,3,3,3,4))\), as stated explicitly by Adeyemo, Bunnett and Levicán-Santibáñez (2025), who thank W. Bruns for informing them of it; we claim nothing new for it. This is an unrefereed note.